8,499 research outputs found
Bar recursion in classical realisability : dependent choice and continuum hypothesis
This paper is about the bar recursion operator in the context of classical
realizability. After the pioneering work of Berardi, Bezem & Coquand [1], T.
Streicher has shown [10], by means of their bar recursion operator, that the
realizability models of ZF, obtained from usual models of -calculus
(Scott domains, coherent spaces, . . .), satisfy the axiom of dependent choice.
We give a proof of this result, using the tools of classical realizability.
Moreover, we show that these realizability models satisfy the well ordering of
and the continuum hypothesis These formulas are therefore realized
by closed -terms. This allows to obtain programs from proofs of
arithmetical formulas using all these axioms.Comment: 11 page
Applying G\"odel's Dialectica Interpretation to Obtain a Constructive Proof of Higman's Lemma
We use G\"odel's Dialectica interpretation to analyse Nash-Williams' elegant
but non-constructive "minimal bad sequence" proof of Higman's Lemma. The result
is a concise constructive proof of the lemma (for arbitrary decidable
well-quasi-orders) in which Nash-Williams' combinatorial idea is clearly
present, along with an explicit program for finding an embedded pair in
sequences of words.Comment: In Proceedings CL&C 2012, arXiv:1210.289
The strength of countable saturation
We determine the proof-theoretic strength of the principle of countable
saturation in the context of the systems for nonstandard arithmetic introduced
in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the
conclusio
- …